Saddle point escape with zeroth-order information. The problem of escaping saddle points in zeroth-order optimization has been studied less often, and we have already listed all known works comparable to our work in the introduction [1]}, [2]}, [3]}; a more detailed comparison of these works with our results has been provided in the discussion following the statement of our main result theorem:convergencemain. We would like to mention that [4]} also includes a convergence result of \(\tilde{O}\left(\frac{d^{1.5}}{\epsilon ^{4.5}} \right)\)  for the case with noisy function evaluations, which is incomparable to our existing work which focuses on the case with exact function evaluation. In addition, [4]} also makes a subGaussian assumption on the estimator noise, which zeroth-order estimators in our paper do not satisfy. Nonetheless, considering the extension to noisy function evaluations will make for important future work.
